I noticed with Duchess Kate when she left the same day. Super women they are! In the US, the usually let woman who went the natural route stay over night, and the woman who did c-section stay up to 3 days.
It’s generally best to let new mothers stay in the hospital at least overnight. This isn’t like some routine outpatient procedure, really. Many things can and often do go wrong post-partum with the mother and/or her baby. If you had pre-eclampsia, a C-Section etc. you have to be extra careful.